In `DeltaPQR `,right-angled at Q,X,Y and Z are points on sides `PR, RQ and PQ `respectively such that `QX_|_PR, XY _|_RQ and XZ_|_PQ` . Find the value of XZ If `PZ = 4 cm and ZQ= 9 cm`.

Options

A. 5 cm.
B. 6 cm.
C. 8 cm.
D. 9 cm.

Answer

To solve the problem, we need to find the length of \( XZ \) in triangle \( PQR \) where \( PZ = 4 \) cm and \( ZQ = 9 \) cm.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a right triangle \( PQR \) with a right angle at \( Q \).
- Points \( X, Y, Z \) are located on sides \( PR, RQ, \) and \( PQ \) respectively.
- The lengths given are \( PZ = 4 \) cm and \( ZQ = 9 \) cm.

2. **Use the Property of Similar Triangles**:
- In right triangles, if a perpendicular is drawn from the right angle to the hypotenuse, the segments created on the hypotenuse are proportional to the lengths of the other two sides.
- Here, triangles \( PZX \) and \( QZX \) are similar.

3. **Set Up the Proportionality**:
- From the similarity of triangles, we have:
\[ \frac{XZ}{PZ} = \frac{ZQ}{XZ} \]
- This leads to the equation:
\[ XZ^2 = PZ \cdot ZQ \]

4. **Substitute the Known Values**:
- Substitute \( PZ = 4 \) cm and \( ZQ = 9 \) cm into the equation:
\[ XZ^2 = 4 \cdot 9 \]
\[ XZ^2 = 36 \]

5. **Calculate \( XZ \)**:
- To find \( XZ \), take the square root of both sides:
\[ XZ = \sqrt{36} = 6 \text{ cm} \]

### Final Answer:
The length of \( XZ \) is \( 6 \) cm.

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